Usua Amanam made the decisive interception near midfield with 2:30 to play as the Pac-12 champion Cardinal (12-2) ended their four-decade drought in the Granddaddy of Them All with arguably the biggest bowl win yet during the long-struggling program’s recent renaissance.

‘‘We knew this was going to be a battle, and we wouldn’t expect it any other way,’’ Stanford coach David Shaw said. ‘‘We know it’s going to be tight, it’s going to be close, and we’re going to find a way to win. That’s the way it’s been all year.’’

Stanford clamped down on the Big Ten champion Badgers (8-6), who lost the Rose Bowl in heartbreaking fashion for the third consecutive season. Montee Ball rushed for 100 yards and his FBS-record 83d touchdown, but Wisconsin managed only 82 yards after halftime.

With impressive defense of its own, Wisconsin still stayed in position for an upset in the one-game return of Hall of Fame coach Barry Alvarez, who was back on the Badgers’ sideline in his red sweater-vest seven years after hanging up his whistle.

‘‘This group of kids has been through a lot, and they competed extremely hard against a very high-quality team,’’ Alvarez said. ‘‘We've played three very good football games [at the Rose Bowl]. These guys played hard. In fact, most people would like to get here once. But we just didn’t get it done.’’

When Bret Bielema abruptly left Wisconsin for Arkansas after winning the Big Ten title game, Alvarez agreed to coach his fourth Rose Bowl before handing off his program to new coach Gary Andersen, who met with Alvarez on the field before the game.

But the Badgers’ third straight Rose Bowl appearance ended in much the same way as the last two: With the Wisconsin offense failing to get the late score they desperately needed.

Curt Phillips went 10 for 16 for 83 yards passing and that crucial interception for Wisconsin, and had 64 yards on the ground. Jordan Fredrick caught a short TD pass right before halftime, but no Badgers receiver had more than Jared Abbrederis’s three catches.

And though Ball became the first player to score touchdowns in three Rose Bowls, the powerful back fell short of Ron Dayne’s career Rose Bowl rushing record, swarmed under by waves of tacklers from one of the toughest defenses in the nation.

Kelsey Young rushed for a score on Stanford’s opening possession, and Taylor scored on the second. Wisconsin kept the Cardinal out of the end zone for the final 51 minutes, holding them to 3 points in the second half, but Stanford’s defense didn’t need any more help in the Cardinal’s eighth straight victory.